ADAPTIVE AND ROBUST TECHNIQUES (ART)FOR THERMOACOUSTIC TOMOGRAPHY

By

YAO XIE

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2006

Copyright 2006

by

Yao Xie

This thesis is dedicated to my parents.

ACKNOWLEDGMENTS

Foremost, I would like to thank my advisor, Professor Jian Li, for her support,

encouragement, and guidance in the past two years. Dr. Li provided me the invaluable

opportunity to investigate those fascinating research problems and always showed full

confidence in me. She always encourages and directs me to develop immature ideas

into a full-blown research paper; meanwhile, she sets a high standard for every work

I have done. I wish I could live up to her expectation in my future career. Also,

I am very grateful to my collaborator, Bin Guo, who has helped me a lot with his

rich experience on breast cancer imaging and numerical simulations. I would like

to thank Professor Petre Stoica, at Uppsala University, to teach me think critically

and analyze problems rigorously. His comments and handwritten notes on signal

processing problems have benefited me greatly. Professor Lihong Wang and Dr.

Geng Ku at Texas A&M University, whose suggestions and experimental data have

made this thesis more complete. I would thank all the lab members: Luzhou Xu,

Xiayu Zheng, Xumin Zhu, William Roberts, Hong Xiong, Zhisong Wang, Yijun Sun,

and Yanwei Wang, on myriads of help they have offered me during my two years in

Spectral Analysis Lab. I would like to thank Professor Liuqing Yang and Professor

Kenneth Slatton for serving in my dissertation committee. Thanks go to all my

friends both at University of Florida and elsewhere who made the last two years full

of enjoyment.

This dissertation is dedicated to my parents, for everything they did for me.

iv

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Thermoacoustic Tomography . . . . . . . . . . . . . . . . . . . . . 11.2 Image Reconstruction Algorithms for TAT . . . . . . . . . . . . . . 2

1.2.1 Related Previous Work . . . . . . . . . . . . . . . . . . . . . 21.2.2 Effects of Acoustic Heterogeneity . . . . . . . . . . . . . . . 3

1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 PRELIMINARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Signal Model and Preprocessing . . . . . . . . . . . . . . . . . . . . 72.2 Waveform and Phase Distortions . . . . . . . . . . . . . . . . . . . 9

3 ADAPTIVE AND ROBUST TECHNIQUES (ART) FOR THERMOA-COUSTIC TOMOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1 Step I of ART: Waveform Estimation . . . . . . . . . . . . . . . . . 123.1.1 Standard Capon Beamforming . . . . . . . . . . . . . . . . . 123.1.2 Robust Capon Beamforming . . . . . . . . . . . . . . . . . . 133.1.3 Delay and Sum . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Step II of ART: Peak Searching . . . . . . . . . . . . . . . . . . . . 173.3 Step III of ART: Intensity Calculation . . . . . . . . . . . . . . . . 18

4 NUMERICAL AND EXPERIMENTAL EXAMPLES . . . . . . . . . . . 22

4.1 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 224.1.1 Numerical Simulation Settings . . . . . . . . . . . . . . . . . 224.1.2 Preprocessing and Parameter Choice . . . . . . . . . . . . . 234.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.3.1 One Small Tumor . . . . . . . . . . . . . . . . . . . 254.1.3.2 Two Closely Located Small Tumors . . . . . . . . . 254.1.3.3 One Large Tumor . . . . . . . . . . . . . . . . . . . 25

v

4.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.1 Experimental Settings . . . . . . . . . . . . . . . . . . . . . . 274.2.2 Parameter Choice . . . . . . . . . . . . . . . . . . . . . . . . 274.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2.3.1 Breast Specimen I . . . . . . . . . . . . . . . . . . 284.2.3.2 Breast Specimen II . . . . . . . . . . . . . . . . . . 294.2.3.3 Mouse Brain . . . . . . . . . . . . . . . . . . . . . 294.2.3.4 Effects of Parameters in ART . . . . . . . . . . . . 30

5 CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . 46

5.1 Wideband Signal Processing . . . . . . . . . . . . . . . . . . . . . . 475.2 Image Registration . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

vi

LIST OF FIGURES

Figure page

2–1 (a): A schematic of a 2-D synthetic aperture based TAT scanning system;(b): a typical acoustic pulse recorded by a transducer (for data measuredfrom breast specimen II). . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2–2 The preprocessed signal recorded by three transducers. The dashed redline corresponds to the calculated peak arrival time; the arrows in eachpanel point to the actual peaks. . . . . . . . . . . . . . . . . . . . . . . . 11

4–1 The setting of the experimental TAT imaging system: (a): for breastspecimens [1]; (b): for mouse brain [2]. . . . . . . . . . . . . . . . . . . . 28

4–2 The 2-D breast model in a x-y coordinate system, with a 2 mm - diametertumor present. Different colors correspond to: (a): different permittivityvalues; (b): different sound speeds. . . . . . . . . . . . . . . . . . . . . . 32

4–3 (a): Comparison between the estimated and fitted arrival time delays,for the simulated breast model with one tumor (the curves for two-tumorcase are similar). Histograms of delay differences: (b): simulated breastmodel with one tumor; (c): breast specimen I; (d): breast specimen II;(e): mouse brain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4–4 Reconstructed images based on the 2-D simulated breast model with one2 mm-diameter tumor. (a): DAS-C; (b): DAS-E1; (c) DAS-E2; (d):RCB-E2, with ε = 0.1M ; (e): DAS-P; (f): ART-P, with ε = 0.1M ; (g):DAS-PP; (h): ART-PP, with ε = 0.1M . . . . . . . . . . . . . . . . . . . 34

4–5 Reconstructed images based on the 2-D simulated breast model, withtwo closely located (2mm spacing) small tumors (2 mm in diameter).The parameter used in RCB and ART is ε = 0.1M . The white dotsin (e) and (f) correspond to the actual (center) location of the tumors.(a): DAS-PP; (b): RCB-E2; (c) ART-P; (d): ART-PP, (e): zoom in ofRCB-E2; (f): zoom in of ART-PP. . . . . . . . . . . . . . . . . . . . . . 35

4–6 Reconstructed images based on the 2-D simulated breast model with onelarge tumor (1 cm in diameter). The white circle in the image correspondsto the actual shape of the tumor. (a): DAS-C; (b): DAS-E1; (c) DAS-E2;(d): RCB-E2, with ε = 0.1M ; (e): DAS-P; (f): ART-P, with ε = 0.1M ;(g): DAS-PP; (h): ART-PP, with ε = 0.1M . . . . . . . . . . . . . . . . . 36

vii

4–7 The values of µ used in ART, on dB scale, in the cases of: (a): single 2mm-diameter tumor; (b): two 2 mm spaced 2 mm-diamter tumors; (c): 1cm-diameter tumor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4–8 Reconstructed images for breast specimen I. (a): DAS-C; (b): DAS-E1;(c) DAS-E2; (d): RCB-E2, with ε = 0.5M ; (e): DAS-P; (f): ART-P, withε = 0.5M ; (g): DAS-PP; (h): ART-PP, with ε = 0.5M ; (i): X-ray image;(j): Inverse solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4–9 Reconstructed images for breast specimen II. (a): DAS-C; (b): DAS-E1;(c) DAS-E2; (d): RCB-E2, with ε = 0.5M ; (e): DAS-P; (f): ART-P, withε = 0.5M ; (g): DAS-PP; (h): ART-PP, with ε = 0.5M ; (i): X-ray image;(j): Inverse solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4–10 Reconstructed images for the mouse brain. (a): DAS-C; (b): DAS-E1;(c) DAS-E2; (d): RCB-E2, with ε = 0.6M ; (e): DAS-P; (f): ART-P,with ε = 0.6M ; (g): DAS-PP; (h): ART-PP, with ε = 0.6M ; (i): Inversesolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4–11 Effects of uncertainty parameter ε on ART-PP, with a searching range[-3,3]. (a): ε = 0.7M ; (b): ε = 0.5M ; (c): ε = 0.3M . . . . . . . . . . . . 44

4–12 Effects of the searching range on the DAS-PP images. (a): searchingrange [-20, 20]; (b): searching range: [-40, 40]; (c): searching range [-60,60]; (d): searching range: [-80, 80]. . . . . . . . . . . . . . . . . . . . . . 45

viii

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of the

Requirements for the Degree of Master of Science

ADAPTIVE AND ROBUST TECHNIQUES (ART)FOR THERMOACOUSTIC TOMOGRAPHY

By

Yao Xie

August 2006

Chair: Jian LiMajor Department: Electrical and Computer Engineering

In this thesis, we present Adaptive and Robust Techniques (ART) for thermoa-

coustic tomography (TAT) for breast cancer detection. TAT is an emerging medical

imaging technique that combines the merits of high contrast due to electromagnetic

or laser stimulation and high resolution offered by thermal acoustic imaging. Image

reconstruction via proper signal processing techniques is one of the key problems in

TAT. The current image reconstruction methods used for TAT, such as the widely

used Delay-and-Sum (DAS) approach, are data-independent and suffer from low res-

olution, high sidelobe levels, and poor interference rejection capabilities. Also, the

amplitude and phase distortion problem of thermoacoustic signals due to propagation

in the non-homogeneous media are not properly handled by the existing methods.

The data-adaptive ART can have much better resolution and much better inter-

ference rejection capabilities than their data-independent counterparts. By allowing

certain uncertainties in the parameters, ART can be used to mitigate the amplitude

and phase distortion problems encountered in TAT. Using both simulated and ex-

perimentally measured data, we demonstrate the excellent performance of ART: high

resolution, low sidelobe level, and much better interference rejection capability.

ix

CHAPTER 1INTRODUCTION

1.1 Thermoacoustic Tomography

Thermoacoustic tomography (TAT) is a new technology with great promise in a

wide span of biomedical applications (see, e.g., [1][3][4][5][6]). Its physical basis lies

in the contrast of the radiation absorption rate among different biological tissues.

In TAT, a short electromagnetic pulse (e.g., microwave or laser) is used to il-

luminate the tissue. When the pulse is absorbed by the tissue, the heating results

in expansion which generates acoustic signals. This phenomenon is called the ther-

moacoustic effect. The acoustic signals are recorded by wideband ultrasound trans-

ducer/transducers, which carry the information of tissue absorption properties. From

the recorded thermoacoustic signals, an image of the tissue absorption properties is

reconstructed. Such an image reveals the physiological and pathological status of the

tissue, which has been used in many applications including breast cancer detection.

The cancerous breast tissues have a two to five times higher microwave absorbtion

rate than their surrounding normal breast tissues, which has been attributed to an

increase in the amount of bound water and sodium within malignant cells [7][8][9].

TAT possesses both fine imaging resolution and good spatial contrast proper-

ties. Microwave imaging offers excellent contrast between cancerous and normal

breast tissue, but the long wavelengths provide poor spatial resolution. Conventional

ultrasound imaging has very fine spatial resolution but poor soft tissue contrast.

Microwave-induced TAT takes advantage of both the high contrast of biological tis-

sue in electromagnetic frequency band and the millimeter range spatial resolution due

to the acoustic signals used in image reconstruction. In microwave-induced TAT, the

1

2

acoustic signal is usually below 2 MHz, resulting in the best axial resolution of 1 mm

[10].

The key problem of TAT is how to map the microwave absorption property

distribution from the measured signal, i.e., how to reconstruct the image. One method

is to use the focused ultrasound transducers to localize the thermoacoustic sources

in linear or sector scans and then obtain the images directly from the data, which

is similar to the pulse-echo ultrasonography [11][12]. An alternative method is to

use an array (real or synthetic aperture) of wideband point detectors to acquire

thermoacoustic data and then reconstruct the absorption distribution. The later

approach provides more flexibility to image reconstruction, which will be considered

in this dissertation.

Although this dissertation only deals with tomography rather than 3-D imaging,

the proposed ART methods could be easily extended to the 3-D imaging case as

well. Herein tomography means that we form a cross sectional image of the tested

tissue. In the tomography system, signals from the selected cross section of the tissue

are recorded by focused transducers. One possible way to form a 3-D image of the

tissue is to scan the cross section across different height. Another way is to use a

three dimensional array of transducers to record signals from all heights and then

reconstruct the 3-D image simultaneously.

1.2 Image Reconstruction Algorithms for TAT

1.2.1 Related Previous Work

Developing accurate and robust image reconstruction methods is one of the key

challenges in TAT. Various image reconstruction algorithms have been developed for

this purpose. By using Radon transformation on the TAT data function, reflectivity

tomography reconstruction algorithms can be used for TAT image reconstruction [13].

Exact inverse solutions have been found for different scanning geometries in both the

frequency domain [14][15] and the time domain [16][17]. Approximate reconstruction

3

algorithms, such as the time-domain Delay-and-Sum (DAS) beamforming method

[18][19] and the optimal statistical approach [20], have also been proposed.

Time-domain approximate reconstruction algorithms such as the DAS type of

data-independent approaches is widely used in medical imaging. In [19], un-weighted

DAS is employed; in [18], the weights for DAS are chosen to be the transducer direc-

tivity to maximize the output Signal to Noise Ratio (SNR). The DAS type approaches

need little prior information on the tissue for image reconstruction and can be fast

and simple to implement to process the wideband acoustic signals. Although not

based on the exact solution, they provide similar image qualities to those of the exact

reconstruction algorithms.

However, these data-independent methods, including DAS, tend to suffer from

poor resolution and high sidelobe level problems. Data-adaptive approaches, such

as the recently introduced Robust Capon Beamforming (RCB) method [21][22], can

have much better resolution and much better interference rejection capability than

their data-independent counterparts.

1.2.2 Effects of Acoustic Heterogeneity

A common assumption of these existing methods is that the surrounding tissue

is acoustically homogeneous. This approximation is inadequate in many medical

imaging applications. According to previous studies, the sound speed in human female

breast varies widely from 1430 m/s to 1570 m/s around the commonly assumed speed

of 1510 m/s [23][24]. The heterogeneous acoustic properties of biological tissues cause

amplitude and phase distortions in the recorded acoustic signals, which can result in

significant degradations in image quality [25].

In Ultra-sound Tomography (UT), wavefront distortion due to heterogeneity of

biological tissue has been studied extensively. Measurements of ultrasonic pulse am-

plitude and arrival time distortion have been obtained for abdominal wall [26] and

human breast [27][28]. Various wavefront correction methods have been proposed

4

[29][30][31]. For example, in [26], a method based on the cross-correlation functions

of the signals recorded by neighboring transducers is used to estimate the wave-

front distortions and make corresponding correction. A more robust but complicated

method, using cross correlation functions with respect to a reference waveform, which

obtained from a group of recorded signals with high similarity, have also been pro-

posed [31]. Various criterions for wavefront correction algorithms have been consid-

ered, including the mean-square error [30], and the speckle brightness [29]. A more

recent approach was designed toward inverse filtering [32]. Some of the aforemen-

tioned methods could be borrowed for the purpose of phase distortion correction in

thermoacoustic signals. However, they are not highly effective at correcting severe

amplitude distortions [32][33], and they usually involve complicated procedures.

Also, the problem in TAT is somewhat different from that in UT. In the breast

UT, the amplitude distortion caused by refraction is more problematic than the phase

distortion induced by acoustic speed variation. In TAT, however, even for the bio-

logical tissue, such as the breast tissue, with a relatively weak heterogeneity, phase

distortion dominates amplitude distortion [25]. This is due to the following reasons.

First, thermoacoustic signals are mostly in a lower frequency range (usually below 1.5

MHz [10]) than those in UT. In the frequency range of TAT, ultrasound scattering is

weak [24]. Second, in TAT, the acoustic signals are induced by electromagnetic ab-

sorption hence only have one-way propagation distortion due to transmission. In UT,

either in the pulse-echo mode or in the transmission mode, two ways of distortions

happen on the wave, during both the transmission and reception propagation. These

unique features suggest that new adaptive and robust imaging techniques should be

designed especially for TAT.

1.3 Contributions

In this thesis, we present new Adaptive and Robust Techniques (ART) based

on RCB for TAT. ART can be used to mitigate the amplitude and phase distortion

5

problems in TAT by allowing certain uncertainties. Specifically, in the first step

of ART, RCB is used for waveform estimation by treating the amplitude distortion

with an uncertainty parameter. In the second step of ART, a simple yet effective

peak searching method is used for phase distortion correction. Compared with other

energy or amplitude based response intensity estimation methods, peak searching

can be used to improve image quality with little additional computational costs.

Moreover, since the acoustic pulse is usually bipolar: a positive peak, corresponding

to the compression pulse, and a negative peak, corresponding to the rarefaction pulse

[18], we can further enhance the image contrast in TAT by using the peak-to-peak

difference as the response intensity for a focal point.

We will demonstrate the excellent performance of ART by using both data sim-

ulated on a 2-D breast model and data experimentally measured from mastectomy

specimens and mouse brain. The numerical examples are designed to test the ability

of ART to detect small tumor, to resolve closely located small tumors, and to form

image of a large tumor with consistent size and shape. The experimental examples

test the performance of ART in real biological tissues, and study the effects of choice

of parameter in ART.

1.4 Dissertation Outline

In Chapter 2, we introduce the data model for a TAT imaging system. First, we

preprocess the signals to compensate for the propagation loss, and time-shift to align

all the signals from a focal point for further processing. The image reconstruction

problem is formulated based on the preprocessed data. A brief review for the causes

of the amplitude and phase distortions of the acoustic signals, and their effects on

the reconstructed images is also given.

Chapter 3 presents the Adaptive and Robust Techniques (ART) for TAT. ART

processes the wideband thermoacoustic signals in time domain. ART consists of three

steps, waveform estimation, peak searching, and response intensity calculation. In the

6

first step, Robust Capon Beamforming (RCB) [21] is used for waveform estimation,

and compared with the conventional Delay-and-Sum (DAS) approach. Step II of ART

introduces peak searching which can effective mitigate the effect of phase distortion.

In Step III, two types of response intensity calculation approaches, the energy based

and amplitude based, are summarized and compared. We also present two new

response intensity calculation methods based on the peak-searching results, which

show advantages compared with the existing methods.

In Chapter 4, we presents some examples to study the performance of ART. The

examples are based on both simulated and real-world experimental data. Compared

with the data-independent and non-robust counterparts, ART has lower sidelobe

level, higher resolution, and much better capability in rejecting interferences. Using

the data from a simulated 2-D breast model, we show that ART can detect small

tumor at accurate location, form an image of the large tumor with consistent shape

and size, and resolve two closely located small tumors. Using the data experimentally

obtained from a mastectomy breast specimen and mouse brain, we show that ART

has meritorious performance in the real scenario. Also, the choice of parameters in

ART are studied.

The conclusions and future work are given in Chapter 5.

CHAPTER 2PRELIMINARY

2.1 Signal Model and Preprocessing

Consider a TAT imaging system as shown in Figure 2–1(a). A stimulating elec-

tromagnetic (laser or microwave) pulse is absorbed by the biological tissue under

testing, which causes a sudden heat change (on the order of 10−4 degrees Celsius

[34]). Due to the thermoacoustic effect, an acoustic pulse is generated which can

be recorded by an ultrasonic transducer array. The transducer array may be a real

aperture array or a synthetic aperture array formed by rotating a sensor around the

tissue and record the acoustic waves at different locations. We assume that the num-

ber of transducers in the array (or in the synthetic aperture array case, the number

of transducer data acquisition locations) is M . Each transducer is assumed to be

omnidirectional; mutual couplings among the transducers are not considered in our

model but the induced errors can be tolerated by our robust algorithms to a certain

extent. The recorded acoustic signals are sufficiently sampled and digitized and a

typical recorded pulse is shown in Figure 2–1(b) (base on the data measured on the

breast specimen II described in Section 6). The data model for the sampled and

digitized acoustic signal recorded by the mth transducer is given by:

xm(n) = sm(n) + em(n), m = 1, · · · ,M. (2–1)

where n is the discrete time index, starting from t0 after excitation pulse. The

scalar sm(n) denotes the signal component, which corresponds to the acoustic pulse

generated at a focal point, and em(n) is the residual term, which includes unmodelled

noise and interference (caused by other sources within the tissue).

7

8

r m+1

r m r

x

y

Tissue

m th scanning stop of transducer

Burried object 0 10 20 30 40 50−3

−2

−1

0

1

2

3

4

5x 10

−3

Time (µ s)

(a) (b)

Figure 2–1: (a): A schematic of a 2-D synthetic aperture based TAT scanning system;(b): a typical acoustic pulse recorded by a transducer (for data measured from breastspecimen II).

The goal of ART is to reconstruct an image of thermoacoustic response intensity

I(r), which is directly related to the absorption property of the tissue, from the

recorded data set {xm(n)}. Herein the (2-D or 3-D) vector r denotes the focal point

location coordinate. To form an image, we scan the focal point at location r to cover

the entire cross-section of the tissue. We allow certain uncertainties in ART to deal

with amplitude and phase distortions caused by the background heterogeneity.

The discrete arrival time of the pulse (for the mth transducer) can be determined

approximately as:

tm(r) =

⌊− t0

∆t+‖r− rm‖

∆tv0

⌋. (2–2)

We will omit the dependence of the arrival time tm(r) on r hereafter for notational

simplicity. Here ∆t is the sampling interval, and the 3-D vector rm denotes the

location of the mth transducer. The sound speed v0 is chosen to be the average

sound speed of the biological tissue under interrogation. The notation ‖x‖ denotes

the Euclidean norm of x, and byc stands for rounding to the greatest integer less than

y. The second term in (2–2) represents the time-of-flight between the focal point and

the mth transducer.

9

The signal components {sm(n)}Mm=1 are approximately scaled and shifted versions

of a nominal waveform s(t) at the source:

sm(n) ≈ exp (−α‖r− rm‖)‖r− rm‖ · s(n− tm), (2–3)

where α is the attenuation coefficient in Nepers/m. In TAT, the major frequency

components of the acoustic signals take a relatively narrow band, and are usually

lower than those in UT [25]. Hence we can approximate α as a frequency independent

constant.

We preprocess the data to time delay all the signals from the focal point r and

compensate for the loss in amplitude due to propagation decay. Let ym(n) denote the

signal after preprocessing to backpropagate the detected signal to the source:

ym(n) = exp (α‖r− rm‖) · ‖r− rm‖xm(n + tm); (2–4)

then the received vector data model can be written as:

y(n) = a0s(n) + e(n), n = −N, · · · , N, (2–5)

where a0 is the corresponding steering vector, which is approximately equal to a =

[1, · · · , 1]T , y(n) = [y1(n), · · · , yM(n)]T , e(n) represents the noise and interference

term after preprocessing, and (·)T denotes the transpose. Here we define the time

interval of interests for the signal y(t) to be from −N to N , which means that we

only take N samples before and after the approximate arrival time given in (2–2)

for the focal point at r. The value of N should be chosen large enough so that the

interval from −N to N covers the expected signal duration in the region of interest.

2.2 Waveform and Phase Distortions

In reality, both the amplitude and the phase (or pulse arrival time) of the acoustic

pulse will be distorted. A major cause for these distortions is the acoustically hetero-

geneous background. Amplitude distortion is mainly due to the interferences caused

10

by multi-path, which is inevitable in the heterogeneous medium: refraction occurs

due to acoustic speed mismatch across the tissue interface; consequently, acoustic

pulses arrived at the transducer will be via different routes and interfere with each

other.

On the other hand, phase distortion is mainly caused by the nonuniform sound

speed. For example, in human female breast the sound speed can vary from 1430 m/s

to 1570 m/s; therefore the actual arrival time will fluctuate around the approximately

calculated time given in (2–2). Moreover, an inaccurate estimate of t0 (t0 is aligned

with the focal point’s signal arrival time) and the transducer calibration error may

also contribute to the phase distortion.

One example for amplitude and phase distortion is shown in Figure 2–2. The

signals are obtained from the second breast specimen in later experimental examples.

The preprocessed signals recorded by three transducers are shown in Figure 2–2,

where the dashed line corresponds the calculated arrival time of the peaks, and the

arrows in each panel point to the actual peaks. Clearly the expected and the actual

peak arrival time do not coincide; their differences are the phase distortion. The shape

of the peaks are different, which is part of the amplitude distortion. Amplitude and

phase distortion will blur the image, raise the image background noise level, lower the

values of the object of interest, and consequently decrease the image contrast [25].

We mitigate the effects of these distortions by allowing a0 to belong to an un-

certainty set centered at a and by considering the signal arriving within the interval

from −N to N . Our ART algorithm consists of three steps: Step I, Robust Capon

Beamforming (RCB) [21][22] for robust waveform estimation; Step II, peak-searching

for phase aberration mitigation; Step III, intensity calculation for forming the final

images.

11

50 100 150 200 250 300−5

0

5x 10

−3

50 100 150 200 250 300−5

0

5x 10

−3

50 100 150 200 250 300−5

0

5x 10

−3

Time Index (Sample)

Tran

duce

r 2

Tran

duce

r 3

Tran

duce

r 1

Figure 2–2: The preprocessed signal recorded by three transducers. The dashed redline corresponds to the calculated peak arrival time; the arrows in each panel pointto the actual peaks.

CHAPTER 3ADAPTIVE AND ROBUST TECHNIQUES (ART) FOR THERMOACOUSTIC

TOMOGRAPHY

3.1 Step I of ART: Waveform Estimation

The first step of ART is to estimate the waveform of the acoustic pulse generated

by the focal point at location r, based on the data model in (2–5). Covariance fitting

based RCB [35] is used to first estimate the steering vector a0, and use the estimated

a0 to obtain an optimal beamformer weight vector for pulse waveform estimation.

It will appear that we have neglected the presence of phase distortion by using

this data model in the first step. However, by allowing a0 to be uncertain, we can

tolerate some phase distortions as well. This approximation causes little performance

degradation to our robust algorithm.

3.1.1 Standard Capon Beamforming

The Capon beamformer [36] is a widely used data-adaptive beamforming method,

which enjoys the advantages of better resolution, lower sidelobe level, and much

better interference rejection capability than the data-independent methods. Capon

beamformer selects the weight vector for waveform estimation adaptive according to

the incoming data, to minimize the output power subject to the linear constraint that

the signal-of-interest does not suffer from any distortion.

The common formulation of Capon is to determine a M × 1 weight vector w to

the following linearly constrained quadratic problem:

minw

w∗Rw subject to w∗a0 = 1, (3–1)

12

13

where

R =1

2N + 1

N∑n=−N

y(n)yT (n) (3–2)

is the sample covariance matrix. Then the weight vector w is used to estimate signal

waveform. The solution to (3–1) is easily derive:

w0 =R−1a0

a∗0R−1a0

. (3–3)

The Capon beamforming can also be reformulated into a covariance fitting form

[35]:

maxσ2

σ2 subject to R− σ2a0a∗0 ≥ 0. (3–4)

The covariance fitting means that herein given R and a0 we wish to determine the

largest possible signal of interest term, σ2a0a∗0, that can be a part of R under the

natural constraint that the noise covariance matrix is positive semi-definite.

The inherent problem of Capon is that, when the knowledge of the SOI steer-

ing vector is imprecise, it will suppress the signal-of-interest as a interference, which

leads to a significant performance degradation. However, the steering vector mis-

match is frequently encountered in practice, due to the looking angle direction errors,

array calibration imperfection, source spreading and local scattering, as well as source

wavefront distortions caused by the environment inhomogeneity [37].

3.1.2 Robust Capon Beamforming

Robust Capon Beamforming (RCB) [21][22][35] is one of most efficient methods

proposed to solve the steering vector mismatch problem in Capon. By assuming that

the true steering vector lies in the vicinity of the nominal steering vector a, RCB

14

consider the following optimization problem [21][22]:

maxσ2,a0

σ2 subject to R− σ2a0aT0 º 0,

‖a0 − a‖2 ≤ ε, (3–5)

where A º 0 means that the matrix A is positive semi-definite, σ2 is the power of the

signal of interest. The second constraint in (3–5) is a spherical uncertainty set; an

elliptical uncertainty set can be used instead if a tighter constraint is desirable [35].

The parameter ε in (3–5) determines the size of the uncertainty set and is a user

parameter. To avoid the trivial solution of a0 = 0, we require that

ε < ‖a‖2. (3–6)

It can be verified that the smaller the ε, the higher the resolution and the stronger

the ability of RCB to suppress an interference that is close to the signal of interest,

and that the larger the ε, the more robust RCB will be to tolerate distortions and

small sample size problems caused by calculating R in (3–2) from a finite number of

data vectors or snapshots. When ε is close to M , RCB will perform like DAS. To

attain high resolution and to effectively suppress interference, ε should be made as

small as possible. On the other hand, the smaller the sample size N or the larger the

distortions, the larger should ε be chosen [21][22]. Since the performance of RCB does

not depend very critically on the choice of ε (as long as it is set to be a “reasonable

value”) [35], such qualitative guidelines are usually sufficient for making a choice of

ε. We will investigate the effect of ε in Section 4. In our examples in Section 4,

we choose certain reasonable initial values for ε, and then make some adjustments

empirically based on image quality: making it smaller when the resulting images have

low resolution, or making it larger when the image is distorted by interferences.

15

By using the Lagrange multiplier method, the solution to (3–5) is given by

[21][22]:

a0 = a−[I + µR

]−1

a, (3–7)

where I is the identity matrix, µ ≥ 0 is the corresponding Lagrange multiplier that

can be solved from the following equation:

∥∥∥(I + µR)−1a∥∥∥

2

= ε. (3–8)

Consider the eigendecomposition on the sample covariance matrix R:

R = UΓUT , (3–9)

where the columns of U are the eigenvectors of R and the diagonal matrix Γ consists

of the corresponding eigenvalues γ1 ≥ γ2 ≥ · · · ≥ γM . Let b = UT a, where bm

denotes its mth element. Then (3–8) can be rewritten as

L(µ) =M∑

m=1

|bm|2(1 + µγm)2

= ε. (3–10)

Note that L(µ) is a monotonically decreasing function of µ, with L(0) > ε by (3−−6)

and limµ→∞ L(µ) = 0 < ε, which means that µ can be solved efficiently, say, by using

the Newton’s method (see [21] for more details). After obtaining the value of µ, the

estimate a0 of the actual steering vector a0 is determined by (3–7).

Observe that there is a “scaling ambiguity” in (3–5) by treating both the signal

power σ2 and the steering vector a0 as unknowns (see [21][35]). The ambiguity exists

in the sense that (σ2, a0) and (σ2/c, c1/2a0) (for any constant c > 0) yields the same

term σ2a0aT0 . To eliminate this ambiguity, we scale the solution a0 to make its norm

satisfy the following condition:

‖a0‖2 = M. (3–11)

16

To obtain an estimate for the signal waveform s(n), we apply a weight vector to

the preprocessed signals {y(n)}Nn=−N . The weight vector is determined by using the

estimated steering vector a0 in the weight vector expression of the standard Capon

beamformer (3–3):

wRCB =‖a0‖M1/2

·

[R + 1

µI]−1

a0

aT0

[R + 1

µI]−1

R[R + 1

µI]−1

a0

. (3–12)

Note that (3–12) has a diagonal loading form, which allows the sample covariance

matrix to be rank-deficient. The beamformer output can be written as:

sRCB(n) = wTRCBy(n), n = −N, · · · , N, (3–13)

which is the waveform estimate for the acoustic pulse generated at the focal point at

location r.

3.1.3 Delay and Sum

RCB can provide a much better waveform estimate than the conventional DAS

but at a higher computational cost. For a single focal point, RCB requires O(M3)

flops, which mainly comes from the eigen-decomposition of the sample covariance

matrix R [21][22]; DAS needs only O(M) flops. DAS can be used as a fast image

reconstruction method to provide initial imaging results.

The weight vector used by (unweighed) DAS for waveform estimation is

wDAS = a, (3–14)

and the estimated waveform is given by

sDAS(n) = wTDASy(n) =

M∑m=1

ym(n), n = −N, · · · , N. (3–15)

17

The weighted DAS [18] uses the transducer directivity rather than the uniform 1/M

as the weights. Since the weights depend on transducer directivity and need to be

measured in practice, we will not consider it herein.

3.2 Step II of ART: Peak Searching

Based on the estimated waveform obtained in Step I for the focal point at loca-

tion r, in Step II of ART, we will search for the two peaks of the bipolar acoustic pulse

generated by the focal point. In a homogeneous background, where phase distortion

is absent, we can accurately calculate the arrival time of the acoustic pulse generated

by the focal point at location r by using (2–2). However, this is never true in hetero-

geneous biological tissues due to the nonuniform sound speed. It was reported in [25]

that when the heterogeneity is weak, such as in the breast tissue, amplitude distortion

caused by multi-path is not severe. We can assume that the original peak remains a

peak in the waveform estimated from Step I of ART. Searching is performed within

an interval around the calculated arrival time. The bipolar acoustic pulse has one

peak positive and another negative. We determine the positive and negative peak

values as follows:

P+ = max

{max

n∈[−∆,∆]s(n), 0

}, (3–16)

P− = min

{min

n∈[−∆,∆]s(n), 0

}, (3–17)

where the searching range [−∆, ∆] ∈ [−N,N ] is around the calculated arrival time

given by (2–2). Here ∆ is a user parameter. Since the peak searching is independent

of the particular waveform estimation methods, we use s(n) to denote the waveform

estimated by either DAS or ART.

The search range is determined by the difference between the true arrival time

tm and the calculated arrival time tm based on (2–2). This arrival time difference has

been analyzed for breast tissue by taking into account its relatively weak heterogeneity

18

acoustic property [25]. In [25], an expression for this difference is given by:

δm(r’) = tm − tm ∝ [v(r’)− v0]

v0

, (3–18)

where r’ is a point within the line connecting the focal point at location r and the mth

transducer at location rm, and v(r’) is the local sound speed. In (3–18), the higher

order terms of [v(r’)− v0]/v0 have been ignored. It is reasonable to assume that v(r’)

is Gaussian distributed with mean v0 and variance σ2v . Consequently the arrival time

difference is also Gaussian distributed with zero-mean and variance σ2δ ∝ σ2

v/v20. If

we choose ∆ = σδ, and the duration of the acoustic pulse is τ , we can find the two

peaks of the pulse within the interval (−σδ, σδ +τ) on the recorded signals with a high

probability of 0.6826. This analysis is consistent with the experimental measurements

in [27][28]. From our examples, we found that a symmetric range [−∆, ∆] around

the estimated arrival time performs similarly to the asymmetric range [−∆, ∆ + τ ],

and we use the former since it is easy to handle in practice. Also, we can use similar

techniques as those in [27][28] to estimate σδ to find a good searching range for Step

II of ART, and to estimate τ for the energy type methods, as shown in our examples

later.

There is a tradeoff in choosing the searching range. The larger the searching

range, the higher the probability we can find the peaks of the acoustic pulse within

the range. However, if the range is chosen too large, the interferences may cause false

peaks, and as a consequence, we are more likely to find a false peak. In our examples

in Section 6, we choose the best searching range empirically based on the estimated

variance of the arrival time difference σδ.

3.3 Step III of ART: Intensity Calculation

After estimating the waveform generated by the focal point at location r, we

need to obtain the response intensity based on the estimated waveform. For the same

19

estimated waveform, different approaches can be taken to evaluate the focal point re-

sponse intensity. These approaches extract different information from the estimated

waveform as the response intensity. The seemingly subtle differences among the inten-

sity calculation methods can result in dramatic changes in the reconstructed image

appearances. These different appearances may be useful to physicians in different

ways.

There are two major types of response intensity measurement approaches: am-

plitude based and energy based. The waveform peak values obtained in Step II of

ART can be used for both approaches.

Conventional DAS uses the amplitude based measure for TAT imaging [18][19][38],

with the corresponding response intensity given by s(0), or equivalently:

IC = s(0) =M∑

m=1

ym(0), (3–19)

where the subscript “C” stands for “Conventional.”

The energy based measure, such as the one used in [38], calculates the response

intensity as follows:

IE1 = s2(0) =

[M∑

m=1

ym(0)

]2

, (3–20)

where the subscript “E1” means “Energy-type 1.”

The entire pulse energy has also been used as an intensity measure, such as

in the mono-static and multi-static microwave imaging for breast cancer detection

[39][40][41], and the intensity is given by:

IE2 =τ∑

n=0

s2(n) =τ∑

n=0

[M∑

m=1

ym(n)

]2

, (3–21)

where the subscript “E2” stands for “Energy-type 2.”

20

We can consider using the peak value as the response intensity measure due to

the bipolar nature of the response at the focal point:

IP =

P+ if |P+| ≥ |P−|;P− otherwise,

(3–22)

where the subscript “P” stands for “Peak,” with P+ and P− defined in (3–16) and

(3–17), respectively. Herein we keep the sign of the maximum amplitude since the

sign of the peak may also contain some information about the focal point.

Peak-searching maximizes the output signal-to-noise ratio. An intuitive expla-

nation is that, given the fact that the acoustic pulse is bipolar [18], if we assume that

the residual term e(t) is stationary, or its power is uniform over time, then the signal-

to-noise ratio is maximized at the (positive or negative) peak of the acoustic pulse.

As a comparison, the conventional DAS (3–19) fixes the samples to be summed up at

the calculated arrival time. Due to phase distortions, the waveform at the calculated

time may be far from the peak value.

We can also employ peak-to-peak difference as the response intensity for the

focal point at location r:

IPP = P+ − P− ≥ 0, (3–23)

where the subscript “PP” denotes the “Peak-to-Peak difference.” Peak-to-peak differ-

ence has higher imaging contrast than peak value measure: the peak-to-peak differ-

ence of the bipolar pulse is approximately twice the absolute peak value, which means

that the output signal power of the former is four times of the latter; yet the noise

power of the former is only twice of the latter. Therefore the output Signal-to-Noise

Ratio (SNR) is doubled by using the peak-to-peak difference rather than the peak

value.

21

Both peak-value and peak-to-peak difference measures belong to the amplitude

based measures.

CHAPTER 4NUMERICAL AND EXPERIMENTAL EXAMPLES

We demonstrate the performance of ART using both numerically simulated and

experimentally measured 2-D TAT data. The ART images are compared with the

DAS images.

4.1 Numerical Examples

4.1.1 Numerical Simulation Settings

We simulate a 2-D cylindrical breast model using the Finite-Difference Time-

Domain (FDTD) [42] method. The 2-D breast model includes 2 mm thick skin, chest

wall, as well as randomly distributed fatty breast tissues and glandular tissues. The

cross-section of the breast model is a half circle with a 10 cm diameter. In the first

numerical example, a 2 mm-diameter tumor is located at 2.2 cm below the skin (at

x = 7.0 cm, y = 6.0 cm). Figure 4–2 shows the shape, dielectric properties and sound

speed variations of the breast model, as well as tumor size and location for the first

example. In the second numerical example, two 2 mm-diameter tumors are located

at x = 13.0 cm, y = 12.1 cm, and at x = 12.4 cm, y = 11.5 cm, with only a 2-mm

spacing between them. In our third example, one large tumor (1 cm in diameter) is

located at x = 12 cm, y = 15 cm. Other properties of the breast model for the second

and third examples are the same as those for the first example.

To reduce the reflections from the skin, the breast model is immersed in a lossless

liquid with permittivity similar to that of the breast fatty tissue. Seventeen trans-

ducers are located on a half circle 10 mm away from the skin to form a real aperture

array.

22

23

The dielectric properties of the breast tissues are assumed to be Gaussian random

variables with variations of ±10% around their nominal values. This variation repre-

sents the upper bound reported in the literature. The nominal values are chosen to

be typical of those reported in the literature [7][8][43][44][45], which is given in Table

1 [41]. The dielectric constants of glandular tissues are between εr = 11 and εr = 15.

The dispersive properties of the fatty breast tissue and those of the tumor are also

considered in the model. The randomly distributed fat breast tissues and glandular

tissues with variable dielectric properties are representative of the non-homogeneity

of the breast of an actual patient.

Following the report that the breast tissues have a weak acoustic heterogeneity

[25], we model the sound speed within the breast as a Gaussian random variable

with variation ±5% around the assumed average sound speed of 1500 m/s. Since the

attenuation coefficient α in (2–3) is small for breast tissue (0.75 dB/MHz/cm) [24]

and the acoustic signals are below 2 MHz, we neglect the exponential attenuation in

acoustic wave propagation. Also, since the acoustic pressure field generated by the

thermoacoustic effect is usually small [34], we do not consider the non-linear acoustic

effects. The differential equations in [16] are used to characterize the thermoacoustic

effect in our FDTD simulations. The probing microwave pulse used here is a mod-

ulated rectangular pulse with a modulating frequency of 800 MHz. The duration of

the pulse is 1 µs.

In the following all the images are displayed on a linear scale, and we will name

the imaging methods by their waveform estimation method followed by the intensity

calculation approach, such as “DAS-C.”

4.1.2 Preprocessing and Parameter Choice

Before applying the aforementioned preprocessing steps and ART, we remove the

strong skin response using techniques similar to those in [38][41]. A calibration signal

is obtained as the average of the recorded signals containing similar skin response.

24

Then the calibration signal is subtracted out from all recorded signals to remove the

skin response as much as possible.

The searching range is chosen by the guidelines presented in Section 4. To obtain

a general profile of the arrival time difference caused by the phase distortion, we use

a simple method similar to the one used in [26]. First, the cross-correlation functions

for all the signals recorded by the two neighboring transducers are obtained. The

peak value of the cross-correlation function is used to estimate the arrival time delay

between the signals recorded by the neighboring transducers. Second, these arrival

time delays are fitted using a fourth-order polynomial curve, which is dominated

by the arrival time delays due to the path length differences in the absent of phase

distortions. The fourth-order polynomial is used since the delay caused by path length

difference should vary smoothly [26]. Figure 4–3(a) shows the estimated arrival time

delay and the delay based on curve fitting. Third, the delay difference between the

estimated arrival time delay and the fitted delay, or the fitting error, is treated as

the arrival time distortion for the transducers. The standard deviation of the delay

difference is used to estimate σδ. Although the accuracy of the cross-correlation

method is limited due to false peaks and jitter problems, it is sufficient to obtain a

qualitative profile for σδ.

Figure 4–3 gives the histogram of the delay difference for all the cases we consid-

ered herein. For our simulated example, the standard deviation of the delay difference

is 4.5, which indicates a weak phase distortion in the breast model. We set an initial

value for ∆ based on the estimated σδ, and adjust the length of the searching range

to achieve the best imaging result.

To estimate the pulse duration τ (used in DAS-E2 and RCB-E2), we select several

typical signals (with clear peaks) and take the average of their pulse durations. In

practice, the acoustic pulse duration is determined by the probing pulse duration,

size and shape of the tumor, as well as the transducer response.

25

4.1.3 Examples

4.1.3.1 One Small Tumor

Figure 4–4 shows the images for the simulated breast model with one 2 mm -

diameter tumor formed using ART and DAS. The tumor response is weak for such

a small tumor. In these images we use ε = 0.1M and the searching range [-14, 14].

Figure 4–4(a) corresponds to DAS-C, where the tumor is buried by interference and

noise. In Figure 4–4(b), DAS-E1 fails to detect the tumor. In Figure 4–4(c), for

DAS-E2, a shadow of the tumor can be seen. In Figure 4–4(d), for RCB-E2, most

of the clutters are cleared up but a strong clutter shows up near the chest wall.

Figures 4–4(e) - 4–4(h) show the results of peak searching; none of them have false

tumors, which may be attributed to proper corrections of phase aberrations. Images

produced by ART-P in Figure 4–4(f) and by ART-PP in Figure 4–4(h) have lower

sidelobe levels and higher resolutions, and the latter has a higher contrast than the

former, due to the latter using the peak-to-peak difference as the intensity measure.

4.1.3.2 Two Closely Located Small Tumors

This example demonstrates the fine resolution of ART. Figure 4–5 shows the

images for the simulated breast model with two 2 mm - diameter tumors, spaced 2

mm away from each other. For this example we use ε = 0.1M and searching range

[-3, 3]. For this case, none of the DAS methods can resolve these two closely located

tumors. Here we just show one DAS image, the DAS-PP image, in Figure 4–5(a).

RCB-E2, in Figure 4–5(b), cannot resolve the tumors either. Both APT-P and APT-

PP can resolve the two tumors, as shown in Figures 4–5(c) and 4–5(d), respectively.

Close examinations of the images for RCB-E2 and ART-PP, shown in Figures 4–5(e)

and 4–5(f), demonstrate the high resolution of ART.

4.1.3.3 One Large Tumor

Figure 4–6 shows the imaging results for the one large tumor (1 cm diameter)

case. Here we set ε = 0.1M and the searching range [-20, 20]. (Note that different

26

tumor sizes and locations will result in different sound speed variations in the breast

model.) For the points inside the tumor sphere, The white circle in the image corre-

sponds to the actual contour of the tumor. Although all the methods can detect the

tumor, only ART can be used to form an image of the tumor with the best agreement

with the actual tumor size and location.

Here the large tumor cannot be assumed a point source, but a 2-D circle. For

a circle of radius rs, zero-crossing points is offset from time 0 by rs/(v0∆t). We

can make the searching range wider to take into this offset into account. In this 1

cm tumor case, the offset can be calculated to be about 10 samples. Note that the

searching range width in our example (empirically decided for a good image quality)

is 40, whereas in the previous 2 mm-diameter tumor case, the searching range width

is 28 (empirically decided for a good image quality). We have increased the searching

range width by 12 samples, which is consistent with our prediction.

Figure 4–7 shows a map of the values of µ used in ART, for each focal point.

Due to the large range of µ, the maps are plotted on dB scale. Note that at the tumor

locations, µ usually takes smaller value than other locations.

We measure the tumor location accuracy by the maximum pixel value in the

image. Also, we define the Signal to Background Ratio (SBR) (i.e., squaring the

pixel value of the image, the ratio of the maxima to total sum of squared value)

as a image quality metric. Such measurements for the images formed by ART are

summarized in Table 2. Note that in all cases ART can detect tumors at their true

locations.

4.2 Experimental Results

We have also tested ART and DAS on three sets of TAT experimental data: two

from mastectomy specimens [1], and another from the mouse brain, obtained by the

Optical Imaging Laboratory at the Texas A&M University.

27

4.2.1 Experimental Settings

The fist two data sets were acquired from mastectomy specimens using a TAT

system as shown in Figure 4–1. Microwave sources were used to heat the specimens

transiently. In the experiment, the breast specimen was formed to a cylindrical shape

inside a plastic bowl. The bowl was immersed in ultrasound coupling medium in a

container. For breast specimen I, the acoustic signals were recorded at 240 equally

spaced scanning stops on a circular track of radius 12.9 cm. The thickness of this

specimen was about 4 cm in a round plastic bowl of 17 cm in diameter. This lesion

was diagnosed as an invasive metaplastic carcinoma with chondroid and squamous

metaplasia. The size of the tumor was measured to be 35 mm in diameter by TAT,

and 36 mm in diameter by radiography (see [1] for details). For breast specimen II,

the scanning radius was 9.7 cm, with 160 scanning stops. This specimen was 9 cm

thick in a round plastic bowl of 11 cm in diameter. The lesion in the specimen was

diagnosed as infiltrating lobular carcinoma; the size of the tumor was about 20 mm

x 12 mm on TAT image, and about 26 mm x 15 mm on the radiography (see [1] for

more details).

The third set of experimental data was obtained from a mouse brain. The data

acquisition procedures are similar to those in [2]. The scanning radius was 3.15 cm,

with 122 scanning stops. But unlike in [2], herein a microwave source was used to

illuminate the mouse brain to obtain a absorption profile different to that in [2].

4.2.2 Parameter Choice

First, we study the delay difference profile for all the breast specimen and mouse

brain data to get a qualitative guide for choosing the searching range in Step II of

ART. The results are shown in Figures 4–3(c) to 4–3(e). Note that breast specimen

II has a larger variance in delay differences than breast specimen I. In Figure 4–3(c),

70% of the delay differences are roughly between -23 to 23 samples, whereas in 4–3(c),

70% of the delay differences are between -40 and 40 samples. Therefore we should set

28

(a) (b)

Figure 4–1: The setting of the experimental TAT imaging system: (a): for breastspecimens [1]; (b): for mouse brain [2].

a larger searching range for breast specimen II than for breast specimen I. In Figure

4–3(d), 70% of the delay difference are between -5 and 5 samples.

4.2.3 Examples

4.2.3.1 Breast Specimen I

Figure 4–8 shows the reconstructed images for breast specimen I. In the following

images, the searching range was set to [-3, 3] after adjustment, and ε = 0.5M for all

the RCB used herein. In Figure 4–8(a), for DAS-C, the dark region shows an blurred

object corresponding to the breast tumor. In Figure 4–8(b), for DAS-E1, the light

region shows a vague boundary of the tumor. Figures 4–8(c), for DAS-E2, and 4–8(d),

for RCB-E2, have similar performances. In Figures 4–8(e), for DAS-P, and 4–8(f), for

ART-P, a dark region with a clear cut has a good correspondence with the location

and shape of the tumor in the radiograph [1]. In Figures 4–8(g), for DAS-PP, and

4–8(h), for ART-PP, not only a clear image of the tumor is obtained, but also the

detailed boundary is revealed. For comparison, the images from X-ray mammography

and the exact inverse solution of TAT (see [1] for more details) are shown in Figures

4–8(i) and 4–8(j), respectively. We give Figure 4–8 and the following Figure 4–9 in

grey scale to have a better comparison with the X-ray images.

29

4.2.3.2 Breast Specimen II

Figure 4–9 shows the reconstructed images for breast specimen II. The tumor

size here is smaller, and a high level of interference and noise is present in the recorded

data. The searching interval is eventually adjust to [-120, 120] and RCB parameter

ε = 0.5M . In Figure 4–9(a), for DAS-C, the true tumor is barely identifiable from the

surrounding clutters. The DAS-E1 shown in Figure 4–9(b) and the DAS-E2 shown in

Figure 4–9(c) provide higher imaging contrast than DAS-C but show strong clutter.

In Figure 4–9(d), for RCB-E2, a false tumor shows up, which demonstrates the need

for robustness in the presence of relatively strong phase distortion. DAS-P is shown in

Figure 4–9(e) and ART-P is shown in Figure 4–9(f). DAS-PP and ART-PP produce

the best images in Figures 4–9(g) and 4–9(h), respectively, with the location and

shape of the tumor consistent with the radiograph in Figure 4–9(i) [1]. If we define

the Signal-to-Background Ratio (SBR) (i.e., squaring the pixel values of the image,

the ratio of the maximum to the total sum of the squared values) as a image quality

measurement metric, ART-PP has an SBR twice that of DAS-PP, which means a 3-

dB gain for ART-PP. For comparison, the image formed by the exact inverse solution

of TAT (see [1] for more details) is shown in Figure 4–9(j).

4.2.3.3 Mouse Brain

The imaging results for the mouse brain are shown in Figure 4–10. In the fol-

lowing images, the searching range was set to [-6, 6] after adjustment, and ε = 0.6M

for all the RCB used herein. Although the interpretations of the images need fur-

ther consideration (for example, with the aid of image registration), we can observe

the improvements of the image quality by using ART-PP. In Figure 4–10(f), ART-P

shows more details, and the image is more clear than that of DAS-P, in Figure 4–10

(e). The energy based energy calculation methods, such as DAS-E2, in 4–10(c), and

RCB-E2, in 4–10(d), tend to blur the details of the image. The images formed by

30

ART-P and ART-PP have the most details and interesting structures. The corre-

spondence of their images with that formed by the exact inverse solution, in Figure

4–10(i), needs further study (see the future work section).

4.2.3.4 Effects of Parameters in ART

The effects of the uncertainty parameter ε in ART is studied in our next example.

We vary ε of RCB used in ART. The imaging results for breast specimen I shown

in Figure 4–11 are consistent with our previous analysis: when ε is large, the perfor-

mance of RCB, in Figure 4–11(a), is close to that of DAS in Figure 4–8(g). When

the parameter ε is small, as shown in Figure 4–11(c), the resolution is improved at

the cost of robustness.

In our last example, the effect of the searching-range width on the imaging qual-

ity is considered. We use DAS-PP as an example since it shows more dependence

on the searching range. The conclusion drawn for DAS applies to ART. A symmet-

ric searching range centered around the calculated arrival time is used. From the

discussions in Section 4, we know that there is a tradeoff in choosing the searching

range. Clearly, when the searching range is too small, such as in Figure 4–12(a),

we miss the true peaks. With an increase in the searching range, the image quality

becomes gradually better as shown in Figures 4–12(b) and 4–12(c). However, when

the searching range passes a certain threshold, with too much interference coming

into the searching range, the image quality degrades because of increased clutters, as

shown in Figure 4–12(d).

From our numerical and experimental examples, we conclude that ART has

higher resolution and better interference rejection capability and more robustness

against wavefront distortion than DAS. Also, we find that the amplitude-based mea-

sures reveal more details of the tumor in the reconstructed images than their energy-

based counterparts. The energy-based measures are not sensitive to phase distortions;

31

however, they tend to blur the reconstructed images, causing lost of details with a

low-pass filter like effect.

32

Table 4–1: Nominal Dielectric Properties of Breast Tissues [41].

Dielectric PropertiesTissuesPermittivity (F/m) Conductivity (S/m)

Immersion Liquid 9 0Chest Wall 50 7Skin 36 4Fatty Breast Tissue 9 0.4Nipple 45 5Glandular Tissue 11-15 0.4-0.5Tumor 50 4

Table 4–2: Various Measurements of Images Formed by ART

Cases Tumor Location (cm × cm) SBR (dB)True location (7.0, 6.0) -

ART-P (6.9, 6.0) -61.0One 2 mm TumorCase

ART-PP (6.9, 6.0) -59.3

True location (7.0, 6.0) and (9.1, 6.9) -ART-P (6.9, 6.0) and (9.1, 6.9) -60.0 and -60.8

Two 2 mm Tu-mors Case

ART-PP (6.9, 6.0) and (9.1, 6.9) -59.9 and -61.0

X (mm)

Y (m

m)

Model: X−Y plane at Z=6cm

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Figure 4–2: The 2-D breast model in a x-y coordinate system, with a 2 mm - diametertumor present. Different colors correspond to: (a): different permittivity values; (b):different sound speeds.

33

0 2 4 6 8 10 12 14 16−40

−30

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e D

elay

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)

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(e)

Figure 4–3: (a): Comparison between the estimated and fitted arrival time delays,for the simulated breast model with one tumor (the curves for two-tumor case aresimilar). Histograms of delay differences: (b): simulated breast model with onetumor; (c): breast specimen I; (d): breast specimen II; (e): mouse brain.

34

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Figure 4–4: Reconstructed images based on the 2-D simulated breast model with one2 mm-diameter tumor. (a): DAS-C; (b): DAS-E1; (c) DAS-E2; (d): RCB-E2, withε = 0.1M ; (e): DAS-P; (f): ART-P, with ε = 0.1M ; (g): DAS-PP; (h): ART-PP,with ε = 0.1M .

35

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(e): Zoom in of (b) (f): Zoom in of (d)

Figure 4–5: Reconstructed images based on the 2-D simulated breast model, withtwo closely located (2mm spacing) small tumors (2 mm in diameter). The parameterused in RCB and ART is ε = 0.1M . The white dots in (e) and (f) correspond to theactual (center) location of the tumors. (a): DAS-PP; (b): RCB-E2; (c) ART-P; (d):ART-PP, (e): zoom in of RCB-E2; (f): zoom in of ART-PP.

36

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Figure 4–6: Reconstructed images based on the 2-D simulated breast model withone large tumor (1 cm in diameter). The white circle in the image corresponds tothe actual shape of the tumor. (a): DAS-C; (b): DAS-E1; (c) DAS-E2; (d): RCB-E2, with ε = 0.1M ; (e): DAS-P; (f): ART-P, with ε = 0.1M ; (g): DAS-PP; (h):ART-PP, with ε = 0.1M .

37

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Figure 4–7: The values of µ used in ART, on dB scale, in the cases of: (a): single 2mm-diameter tumor; (b): two 2 mm spaced 2 mm-diamter tumors; (c): 1 cm-diametertumor.

38

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Figure 4–8: Reconstructed images for breast specimen I. (a): DAS-C; (b): DAS-E1;(c) DAS-E2; (d): RCB-E2, with ε = 0.5M ; (e): DAS-P; (f): ART-P, with ε = 0.5M ;(g): DAS-PP; (h): ART-PP, with ε = 0.5M ; (i): X-ray image; (j): Inverse solution.

40

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Figure 4–9: Reconstructed images for breast specimen II. (a): DAS-C; (b): DAS-E1;(c) DAS-E2; (d): RCB-E2, with ε = 0.5M ; (e): DAS-P; (f): ART-P, with ε = 0.5M ;(g): DAS-PP; (h): ART-PP, with ε = 0.5M ; (i): X-ray image; (j): Inverse solution.

42

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Figure 4–10: Reconstructed images for the mouse brain. (a): DAS-C; (b): DAS-E1;(c) DAS-E2; (d): RCB-E2, with ε = 0.6M ; (e): DAS-P; (f): ART-P, with ε = 0.6M ;(g): DAS-PP; (h): ART-PP, with ε = 0.6M ; (i): Inverse solution.

44

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45

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Figure 4–12: Effects of the searching range on the DAS-PP images. (a): searchingrange [-20, 20]; (b): searching range: [-40, 40]; (c): searching range [-60, 60]; (d):searching range: [-80, 80].

CHAPTER 5CONCLUSIONS AND FUTURE WORK

In this dissertation, we propose new Adaptive and Robust Techniques (ART)

for thermoacoustic tomography, and compare their performances with other existing

methods. ART is robust to the amplitude and phase distortions in the recorded signals

caused by the acoustic heterogeneity of biological tissues. ART consists of three

steps: in the first step, ART uses the data-adaptive Robust Capon Beamforming

(RCB) for waveform estimation; in the second step of ART, a simple yet effective

peak searching method is used to mitigate the phase distortion in the estimated

waveform; in the third step, the response intensity is calculated for the focal point

using various approaches, among which the peak-to-peak difference measure further

enhances the image contrast. The parameters used in ART: the uncertainty set size

and the searching range width, can be determined by using the estimates of arrival

time difference variance, and adjusted according to the imaging quality.

Examples based on a numerically simulated 2-D breast model and three sets

of experimentally measured data from human mastectomy specimens and a mouse

brain demonstrate the excellent performance of ART: high resolution, low-side lobe

level, and much improved interference suppression capability. Also, these examples

demonstrate that ART is promising in detect tumors in the real breast tissues with

accurate location and size, and in brain imaging.

Future work may be to incorporate wideband signal processing techniques in

ART, and to use the image registration techniques to align the images formed by

different methods.

46

47

5.1 Wideband Signal Processing

We could consider using wideband signal model, such as Auto-Regression (AR)

model or Auto-Regression Moving Average (ARMA) model [46], to fit the thermoa-

coustic signals, since they are wideband. The wideband signal processing technique

could be employed. For example, in ART signals are processed in time domain based

on time delay; we could also consider using tap and delay line [47] rather than a

weight vector to estimate the signal waveform. Also, we could consider the frequency

domain wideband signal processing approach: divide the wideband signal signal into

narrowband frequency bins, and then apply incoherent or coherent narrowband pro-

cessing [47]. Wideband signal processing has more degree of freedom hence would be

more robust to signal waveform distortions, but they require higher computational

complexity than the weight vector approach in ART. A good tradeoff could be sought.

5.2 Image Registration

Image registration techniques could be used to align images about one object,

obtained from different imaging methods, or even different imaging modalities. In

our numerical examples, we found that images about one object could take one quite

different appearances. A spatial alignment of the features in different images is often

desired. For example, in the experimental example the mouse brain image recon-

structed by ART-P, in Figure 4–10(f), shows quite different details to that by the

inverse solution, in Figure 4–10(i), but we can observe some correspondence between

the two Figures. Image registration aims to find the correspondence, or to spatially

align images about one object [48].

The registration algorithms could be based on the segmented features, or the

small vortexes (a larger unit than image pixel) [49]. The alignment in image reg-

istration could be rigid, which means only linear image transformation is used, or

non-rigid, which means that non-lineaer image transformation could also be used. A

comprehensive software called MAIMI Fuse [50], developed by Meyer etal., includes

48

many algorithms, feature based or vortex based, rigid (using linear image transforma-

tion) or non-rigid (using non-linear image transformation). In the future, we could

consider using image registration to cross-validate the images formed by different

methods.

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BIOGRAPHICAL SKETCH

Yao Xie was born in Huhhot, Inner Mongolia, People’s Republic of China, in

1982. She received the B.Sc. degree in Electrical Engineering and Information Sci-

ence from the University of Science and Technology of China (USTC), Hefei, China,

in July, 2004, and M.Sc. degree in the Department of Electrical and Computer En-

gineering at University of Florida, Gainesville, Florida, in August, 2006. She is a

member of Tau Beta Pi and Etta Kappa Nu. She was the first place winner in the

Student Best Paper Contest at the 2005 Annual Asilomar Conference on Signals,

Systems, and Computers, for her work on breast cancer detection.

53

I certify that I have read this study and that in my opinion it conforms toacceptable standards of scholarly presentation and is fully adequate, in scope andquality, as a dissertation for the degree of Master of Science.

Jian Li, Chair,Professor of

Electrical and Computer Engineering

I certify that I have read this study and that in my opinion it conforms toacceptable standards of scholarly presentation and is fully adequate, in scope andquality, as a dissertation for the degree of Master of Science.

Liuqing YangAssistant professor of

Electrical and Computer Engineering

I certify that I have read this study and that in my opinion it conforms toacceptable standards of scholarly presentation and is fully adequate, in scope andquality, as a dissertation for the degree of Master of Science.

Kenneth Clint SlattonAssistant Professor of

Electrical and Computer Engineering,and Civil and Coastal Engineering

This dissertation was submitted to the Graduate Faculty of the College of En-gineering and to the Graduate School and was accepted as partial fulfillment of therequirements for the degree of Master of Science.

August 2006Pramod P. KhargonekarDean, College of Engineering

Kenneth GerhardtDean, Graduate School

GROWTH CURVE MODELS IN SIGNAL PROCESSING APPLICATIONS

Yao Xie(352) 871–0826Department of Electrical and Computer EngineeringChair: Jian LiDegree: Master of ScienceGraduation Date: August 2006

Thermoacoustic tomography (TAT) is an emerging medical imaging technique that

has excellent imaging quality: high contrast, in differentiating various tissues, and

high resolution, in rendering details in the image. TAT can be potentially used in a

wide span of medical imaging applications, including early breast cancer detection.

Breast cancer takes tremendous toll to our society. Diagnosis at the earliest stage is

the best hope of surviving breast cancer.

One of the key problems encountered in TAT is the image reconstruction from

TAT signals. The current image reconstruction methods used in TAT, such as the

widely used Delay-and-Sum (DAS) approaches, are data-independent and suffer from

poor imaging quality: low resolution, high sidelobe levels, and inadequate interfer-

ence rejection capabilities. Also, The image quality degradation problem due to the

distortions of TAT signals while propagating in the biological tissues has not been

properly handled in the existing methods.

In this thesis, we present Adaptive and Robust Techniques (ART) for TAT, with

applications in breast cancer detection and mouse brain imaging. The data-adaptive

ART can have much better imaging quality: better resolution, interference rejection

capabilities, than their data-independent counterparts. By allowing certain uncer-

tainties, ART can be used to mitigate the amplitude and phase distortion problems

encountered in TAT. Using both simulated and experimentally measured data, we

demonstrate the excellent performance of ART: high resolution, low sidelobe level,

and much better interference rejection capability.